Properties

Label 2-5850-1.1-c1-0-49
Degree $2$
Conductor $5850$
Sign $-1$
Analytic cond. $46.7124$
Root an. cond. $6.83465$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 4·7-s − 8-s − 4·11-s + 13-s + 4·14-s + 16-s + 4·17-s + 7·19-s + 4·22-s − 4·23-s − 26-s − 4·28-s − 5·29-s + 4·31-s − 32-s − 4·34-s + 9·37-s − 7·38-s + 5·41-s − 10·43-s − 4·44-s + 4·46-s − 3·47-s + 9·49-s + 52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s − 1.20·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.970·17-s + 1.60·19-s + 0.852·22-s − 0.834·23-s − 0.196·26-s − 0.755·28-s − 0.928·29-s + 0.718·31-s − 0.176·32-s − 0.685·34-s + 1.47·37-s − 1.13·38-s + 0.780·41-s − 1.52·43-s − 0.603·44-s + 0.589·46-s − 0.437·47-s + 9/7·49-s + 0.138·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5850\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(46.7124\)
Root analytic conductor: \(6.83465\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5850,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.84839466627882260531660648221, −7.17820801860370475579712412526, −6.36131625563931173652001453945, −5.76401865220602795492935628012, −5.07341508593951813406554916569, −3.72568082704772565766838040552, −3.14004679816178465453479694250, −2.42523891159489288106708860431, −1.07539534286625485909191197391, 0, 1.07539534286625485909191197391, 2.42523891159489288106708860431, 3.14004679816178465453479694250, 3.72568082704772565766838040552, 5.07341508593951813406554916569, 5.76401865220602795492935628012, 6.36131625563931173652001453945, 7.17820801860370475579712412526, 7.84839466627882260531660648221

Graph of the $Z$-function along the critical line